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NEW YORK UNIVERSITY
INSTITUTE CF MATHEMATICAL SCIENCES
25 Wiverly Place, New York 3, N. "«.
NEW YORK UNIVERSITY
INSTITUTE OF MATHEMATICAL SCIENCES
DIVISION OF ELECTROMAGNETIC RESEARCH I
RESEARCH REPORT No. EM-58 !
MULTIPLE SCATTERING OF WAVES
BY PLANAR RANDOM DISTRIBUTIONS
OF PARALLEL CYLINDERS AND BOSSES
by
VIC TWERSKY
CONTRACT No. AF-19(122)-42
OCTOBER 1953
NEW YORK UNIVERSITY
Institute of Mathematical Sciences
Division of Electromagnetic Research
Research Report No. EM-58
MULTIPLE SCATTERING OF WAVES BY PLANAR RANDOM DISTRIBUTIONS
OF PARALLEL CYLINDERS AND BOSSES
by
Vic Twersky
Vi,c 'I'wersky^
Morris Kiine
Project Director
The research reported in this document has been made possible through
support and sponsorship extended by the Air Force Cambridge Research
Center, under Contract No. AF-19(122)-42. It is published for technical
information only, and does not necessarily represent recommendations
or conclusions of the sponsoring agency.
October, 1953
New York, 1953
Manufactured in the United States for New York University Press
by the University's Office of Publications and Printing
IA3LE or CONTEIITS
P8^
Abstract
1, Introduction ^
2, Preliminary Consideration and Statement of the
Problem ^
2.1 Scattering "by a Planar Configuration of
Parallel Cylinders 3
2.2 The Ensemljle of Configurations 9
2.3 Statement of the Problem ^3
3, The Average Wave Function ^^
3.1 Derivation of a Closed- form Approximation
for the Scattering Coefficients 17
3.2 Analysis of the Functions Involved 21
3.3 The Range of Validity of the Closed-fona
Representation 28
U, The Average Intensity 32
5. The Average Energy Flux ^1
6, Distribution of Bosses on a Perfectly Reflecting
Plane
Appendix
Abstract
56
The two-dimensional problem of multiple scattering of a plane
wave by a planar 'random' distribution of parallel cylinders is considered.
A formal solution,> 0, then the roots siaj' he approxi-
Tuated hy v^i^ (2WTi + c)/h. We find e = -i2inTp6, w^ ~ i2wrrp, and
consequently (l^) reduces to
00 iu2npu â–
(16) p(u)^p+^ ^^^iuznph • u>h^h.
tf= +1
We refer to (l6) as the 'more ordered' case. In the limit 6 — > 0, (l6)
reduces to the results for a diffraction grating of spacing "b.
2.3 Stfttftment of the Frohlem
We are now in a position to state explicitly the prohlem
considered in this paper. We are given the wave scattered hy a
single configuration of cylinders, i.e., xy = ^>/-» where vj/^ as
given "by (3) and (U) is expressed in terms of the known solution
for the single cylinder. We seek to average \|r and |\|;| over the
ensemhle specified "by the distrilmtion f-anction
*2^^8» ^s'^ ^ ^l^'^s^ ^^^s'^'s'^ ^°^ pairs where W^ and P are ^iven hy ^10)
and (ll). We also seek closed-form approximations for the averages
which may "be applied to the physical prohlem of scattered reflection
from a striated surface.
Sections 3 ^^^ ^ are concerned with the average values
of >p and |\|f| respectively. Section 5 deals with the average
energy fliix and certain energy theorems. Section 6 treats the
analogous proTslem for "bosses on a perfectly reflecting plane.
In the following sections we use either a "bar or angular
"brackets to indicate an average value.
^Ik-
3» The Average Wave S anction
The average value of the scattered wave defined "by (l),
(3) axid (^) niay he written as
8 3 J J
(17)
= E Z ^"^ / \ ^^8 ^^^ (-i'>Q^3sina)H^(fcr^ )e3p (inO^ ) < B^^>^ dy^
an "^ — d
where < B > indicates that B as in (^) is to he averaged over all
1^ s ns ^^
variahles except y , i»e., over all configurations assumed hy cylinders
s' ^ 8. We refer to < > as an average with one variable held fixed.
(We use essentially the same notation and nomenclature as Foldy and
n
"Lblx • ) Similarly we have
^^, dy^. 1
where < B •> t is the average scattering coefficient with two variables
OS SS '
held fixed.
We now restrict the parameters so that d— »ooand H—> 00 while
HW, (y ) = N/2d = p remains constant. As a consequence we have
(19) =, = B .
ns 3 ns'^s' - n »
-15-
thus the average scattering coefficient with one varia'ble held fixed
is independent of y ; i.e., no 'end effects* occur for an infinite
s
range. (The result (l9), which follows from elementary physical
considerations, can he deduced from (l8) with B -*oo, d -»-oo, if it is kept
in mind that P(y |y ,) as defined in (lo) is a function only of
S 8
|y -7 J.) In view of (l9) we may sum over s in (l?), replace y
hy y', and write
(20) W^pY: vV°°«"''^'""' H^OcrOe^'^^'ay'. r- = [z^ ^(y-y.)^]' = ^ .
n -00
We reserve discussion of B for Section 3.1.
n
The integral over y' can he reduced to one evaluated exactly
12
hy Eeiche :
/CO I 5"
Ejc I 1^2 )
-00
1 + i v
' 1 + V
e-^^ dv
(21)
2 i
â– â– e
1/2 2
Kq - p
- ^n-
-1
INSTITUTE CF MATHEMATICAL SCIENCES
25 Wiverly Place, New York 3, N. "«.
NEW YORK UNIVERSITY
INSTITUTE OF MATHEMATICAL SCIENCES
DIVISION OF ELECTROMAGNETIC RESEARCH I
RESEARCH REPORT No. EM-58 !
MULTIPLE SCATTERING OF WAVES
BY PLANAR RANDOM DISTRIBUTIONS
OF PARALLEL CYLINDERS AND BOSSES
by
VIC TWERSKY
CONTRACT No. AF-19(122)-42
OCTOBER 1953
NEW YORK UNIVERSITY
Institute of Mathematical Sciences
Division of Electromagnetic Research
Research Report No. EM-58
MULTIPLE SCATTERING OF WAVES BY PLANAR RANDOM DISTRIBUTIONS
OF PARALLEL CYLINDERS AND BOSSES
by
Vic Twersky
Vi,c 'I'wersky^
Morris Kiine
Project Director
The research reported in this document has been made possible through
support and sponsorship extended by the Air Force Cambridge Research
Center, under Contract No. AF-19(122)-42. It is published for technical
information only, and does not necessarily represent recommendations
or conclusions of the sponsoring agency.
October, 1953
New York, 1953
Manufactured in the United States for New York University Press
by the University's Office of Publications and Printing
IA3LE or CONTEIITS
P8^
Abstract
1, Introduction ^
2, Preliminary Consideration and Statement of the
Problem ^
2.1 Scattering "by a Planar Configuration of
Parallel Cylinders 3
2.2 The Ensemljle of Configurations 9
2.3 Statement of the Problem ^3
3, The Average Wave Function ^^
3.1 Derivation of a Closed- form Approximation
for the Scattering Coefficients 17
3.2 Analysis of the Functions Involved 21
3.3 The Range of Validity of the Closed-fona
Representation 28
U, The Average Intensity 32
5. The Average Energy Flux ^1
6, Distribution of Bosses on a Perfectly Reflecting
Plane
Appendix
Abstract
56
The two-dimensional problem of multiple scattering of a plane
wave by a planar 'random' distribution of parallel cylinders is considered.
A formal solution,> 0, then the roots siaj' he approxi-
Tuated hy v^i^ (2WTi + c)/h. We find e = -i2inTp6, w^ ~ i2wrrp, and
consequently (l^) reduces to
00 iu2npu â–
(16) p(u)^p+^ ^^^iuznph • u>h^h.
tf= +1
We refer to (l6) as the 'more ordered' case. In the limit 6 — > 0, (l6)
reduces to the results for a diffraction grating of spacing "b.
2.3 Stfttftment of the Frohlem
We are now in a position to state explicitly the prohlem
considered in this paper. We are given the wave scattered hy a
single configuration of cylinders, i.e., xy = ^>/-» where vj/^ as
given "by (3) and (U) is expressed in terms of the known solution
for the single cylinder. We seek to average \|r and |\|;| over the
ensemhle specified "by the distrilmtion f-anction
*2^^8» ^s'^ ^ ^l^'^s^ ^^^s'^'s'^ ^°^ pairs where W^ and P are ^iven hy ^10)
and (ll). We also seek closed-form approximations for the averages
which may "be applied to the physical prohlem of scattered reflection
from a striated surface.
Sections 3 ^^^ ^ are concerned with the average values
of >p and |\|f| respectively. Section 5 deals with the average
energy fliix and certain energy theorems. Section 6 treats the
analogous proTslem for "bosses on a perfectly reflecting plane.
In the following sections we use either a "bar or angular
"brackets to indicate an average value.
^Ik-
3» The Average Wave S anction
The average value of the scattered wave defined "by (l),
(3) axid (^) niay he written as
8 3 J J
(17)
= E Z ^"^ / \ ^^8 ^^^ (-i'>Q^3sina)H^(fcr^ )e3p (inO^ ) < B^^>^ dy^
an "^ — d
where < B > indicates that B as in (^) is to he averaged over all
1^ s ns ^^
variahles except y , i»e., over all configurations assumed hy cylinders
s' ^ 8. We refer to < > as an average with one variable held fixed.
(We use essentially the same notation and nomenclature as Foldy and
n
"Lblx • ) Similarly we have
^^, dy^. 1
where < B •> t is the average scattering coefficient with two variables
OS SS '
held fixed.
We now restrict the parameters so that d— »ooand H—> 00 while
HW, (y ) = N/2d = p remains constant. As a consequence we have
(19) =, = B .
ns 3 ns'^s' - n »
-15-
thus the average scattering coefficient with one varia'ble held fixed
is independent of y ; i.e., no 'end effects* occur for an infinite
s
range. (The result (l9), which follows from elementary physical
considerations, can he deduced from (l8) with B -*oo, d -»-oo, if it is kept
in mind that P(y |y ,) as defined in (lo) is a function only of
S 8
|y -7 J.) In view of (l9) we may sum over s in (l?), replace y
hy y', and write
(20) W^pY: vV°°«"''^'""' H^OcrOe^'^^'ay'. r- = [z^ ^(y-y.)^]' = ^ .
n -00
We reserve discussion of B for Section 3.1.
n
The integral over y' can he reduced to one evaluated exactly
12
hy Eeiche :
/CO I 5"
Ejc I 1^2 )
-00
1 + i v
' 1 + V
e-^^ dv
(21)
2 i
â– â– e
1/2 2
Kq - p
- ^n-
-1
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